A box with a mass of 10 kg is placed on an inclined plane that makes a 60^∘ angle with the horiz

step 1 We have given a box of mass 10 kg is placed on an inclined plane. This angle is 60 degree. So we

A box with a mass of 10 kg is placed on an inclined plane...

A box with a mass of 10 $\mathrm{kg}$ is placed on an inclined plane that makes a $60^{\circ}$ angle with the horizontal. The coefficient of static fraction $\left(\mu_{\mathrm{s}}\right)$ between the box and the inclined plane is $0.2 .$ What force must be applied on the box in order to prevent the box from sliding down the inclined plane? (A) 32.7 $\mathrm{N}$ (B) 48.3 $\mathrm{N}$ (C) 56.2 $\mathrm{N}$ (D) 76.6 $\mathrm{N}$

A box with a mass of 10 $\mathrm{kg}$ is placed on an inclined plane that makes a $60^{\circ}$ angle with the horizontal. The coefficient of static fraction $\left(\mu_{\mathrm{s}}\right)$ between the box and the inclined plane is $0.2 .$ What force must be applied on the box in order to prevent the box from sliding down the inclined plane?
(A) 32.7 $\mathrm{N}$
(B) 48.3 $\mathrm{N}$
(C) 56.2 $\mathrm{N}$
(D) 76.6 $\mathrm{N}$

Key Concepts

Force Decomposition on an Inclined Plane

When dealing with objects on an inclined plane, it is essential to resolve the gravitational force into two components: one parallel to the surface (which encourages sliding down) and one perpendicular to the surface (which is balanced by the normal force). This decomposition allows for an easier analysis of the forces acting on the object and helps in setting up equations for equilibrium.

Static Friction

Static friction is the resistive force that prevents an object from starting to move when a force is applied. Its maximum value is proportional to the normal force between the surfaces and is given by the product of the coefficient of static friction and the normal force. This concept is crucial in determining whether an applied force or gravitational component will cause motion or if static friction can counteract it.

Equilibrium Conditions

In problems involving forces on an inclined plane, equilibrium refers to a state where the net force acting on the object is zero, preventing any acceleration. This means that the sum of all forces in each direction (parallel and perpendicular to the plane) must cancel out. Establishing equilibrium conditions is key to solving for unknown forces, such as the necessary applied force to prevent sliding.

Newton's Second Law

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the context of equilibrium problems, Newton's Second Law simplifies to the condition that the net force is zero, thus guiding the setup of equations used to find the required forces to prevent motion.

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